diff --git a/Algebraic_kernel_d/doc_tex/Curve_and_curve_pair_analysis_2_ref/AlgebraicCurveKernel_2.tex b/Algebraic_kernel_d/doc_tex/Curve_and_curve_pair_analysis_2_ref/AlgebraicCurveKernel_2.tex
index ec20a96ceba..81375bfa579 100644
--- a/Algebraic_kernel_d/doc_tex/Curve_and_curve_pair_analysis_2_ref/AlgebraicCurveKernel_2.tex
+++ b/Algebraic_kernel_d/doc_tex/Curve_and_curve_pair_analysis_2_ref/AlgebraicCurveKernel_2.tex
@@ -1,19 +1,26 @@
-\begin{ccRefConcept}{AlgebraicCurveKernel_2}
+\begin{ccRefConcept}{AlgebraicKernelBase_2}
 
 \ccDefinition
-
-
-The \ccc{AlgebraicCurveKernel_2} concept is meant to provide the curved
-kernel with all the algebraic functionalities on bivariate polynomials
-required for the manipulation of arcs of algebraic curves of general degree
-$\R^2$ using an $y$-per-$x$-view on the curves.
+The \ccc{AlgebraicKernelBase_2} concept is meant to provide the curved
+kernel with very fundamental algebraic functionalities on uni- and bivariate
+polynomials.
 
 \ccTypes
 
-A model of \ccc{AlgebraicCurveKernel_2} is supposed to provide
+A model of \ccc{AlgebraicKernelBase_2} is supposed to provide
 
 \ccNestedType{Coefficient}{A model of \ccc{IntegralDomain}. }
 
+\ccNestedType{Polynomial_1}{A model of \ccc{Polynomial_1}, for
+univariate polynomials when the \ccc{Coefficient} type is an 
+\ccc{IntegralDomain}. It must also be possible to perform \ccc{Canonicalize, 
+    GcdUpToConstantFactor, 
+    IntegralDivUpToConstantFactor, 
+    MakeSquareFreeUpToConstantFactor, and 
+    SquareFreeFactorizationUpToConstantFactor. 
+    Also needed some PolynomialConstruction concept.}
+} 
+
 \ccNestedType{Polynomial_2}{A model of \ccc{Polynomial_2}, for
   bivariate polynomials on an \ccc{IntegralDomain} coefficient type. 
   it must also be possible to perform \ccc{Canonicalize, 
@@ -21,24 +28,17 @@ A model of \ccc{AlgebraicCurveKernel_2} is supposed to provide
     IntegralDivUpToConstantFactor, 
     MakeSquareFreeUpToConstantFactor, and 
     SquareFreeFactorizationUpToConstantFactor} on this type - maybe using a
-  traits class. Also needed some PolynomialConstruction concept
+  traits class. Also needed some PolynomialConstruction concept.
 } 
 
-% TODO: Boundary: RealEmbeddable that is 'dense R'
+\ccNestedType{Solution_1}{A model of
+\ccc{AlgebraicKernelBase_2::Solution_1}, for real roots of univariate 
+polynomials of type \ccc{Polynomial_1}.}
 
 \ccNestedType{Solution_2}{A model of
-\ccc{AlgebraicCurveKernel_2::Solution_2}, for solutions of systems
+\ccc{AlgebraicKernelBase_2::Solution_2}, for solutions of systems
 of two bivariate polynomials of type \ccc{Polynomial_2}.}
 
-\ccNestedType{CurvePairAnalysis_2}{A model of
-\ccc{AlgebraicCurveKernel_2::CurvePairAnalysis_2}, 
-for analysing a pair of curves
-defined as two analyses of type \ccc{CurveAnalysis_2}.}
-
-\ccNestedType{CurveAnalysis_2}{A model of
-\ccc{CurvePairAnalyis_2::CurveAnalysis_2}, for analysing single
-curves defined as bivariate polynomials of type \ccc{Polynomial_2}.}
-
 \ccNestedType{Compare_x_2}{A model of the concept 
 \ccc{AlgebraicKernel_d_2::CompareX_2}.}
 \ccGlue
@@ -49,4 +49,78 @@ curves defined as bivariate polynomials of type \ccc{Polynomial_2}.}
 \ccc{AlgebraicKernel_d_2::CompareXY_2}.}
 
 
+\end{ccRefConcept}
+
+\begin{ccRefConcept}{AlgebraicKernelBasic_2}
+\footnote{Adapted the concepts \ccc{AlgebraicKernel_d_1} and 
+\ccc{AlgebraicKernel_d_2} and re-assigned types/functors 
+to Base and Basic part.}
+
+\ccDefinition
+
+The \ccc{AlgebraicKernelBasic_2} concept is meant to provide the curved
+kernel with all the algebraic functionalities on uni- and bivariate polynomials
+required for the manipulation of arcs of algebraic curves of general degree
+$d$ in $\R^2$.
+
+\ccNestedType{Solve_1}{A model of the concept
+\ccc{AlgebraicKernelBasic_1::Solve_1}.} 
+
+\ccNestedType{Sign_at_1}{A model of the concept
+\ccc{AlgebraicKernelBasic_1::SignAt_1}.}
+
+\ccNestedType{Derivative_1}{A model of the concept
+\ccc{AlgebraicKernelBasic_1::Derivative}.} 
+
+\ccNestedType{Compare_1}{A model of the concept 
+\ccc{AlgebraicKernelBasic_1::Compare_1}.}
+
+\textbf{Question: what is the righ interface with AK 
+for \ccc{compare_y_at_x_right}?}
+
+\ccNestedType{Solve_2}{A model of the concept
+\ccc{AlgebraicKernelBasic_2::Solve_2}.} 
+
+\ccNestedType{Sign_at_2}{A model of the concept
+\ccc{AlgebraicKernelBasic_2::SignAt_2}.}
+
+\ccNestedType{Derivative_x_2}{A model of the concept
+\ccc{AlgebraicKernelBasic_2::DerivativeX_2}.} 
+
+\ccNestedType{Derivative_y_2}{A model of the concept
+\ccc{AlgebraicKernelBasic_2::DerivativeY_2}.} 
+
+\ccNestedType{X_critical_points_2}{A model of the concept 
+\ccc{AlgebraicKernelBasic_2::XCriticalPoints_2}.}
+\ccGlue
+\ccNestedType{Y_critical_points_2}{A model of the concept 
+\ccc{AlgebraicKernelBasic_2::YCriticalPoints_2}.}
+
+\end{ccRefConcept}
+
+\begin{ccRefConcept}{AlgebraicKernelCCPA_2}
+
+\ccDefinition
+
+The \ccc{AlgebraicKernelCCPA_2} concept refines the \ccc{AlgebraicKernelBase_2}
+concept with functionality on bivariate polynomials
+required for the manipulation of arcs of algebraic curves of general degree
+$\R^2$ using an $y$-per-$x$-view on the curves.
+
+\ccRefines
+\ccc{AlgebraicKernelBase_2}
+
+\ccTypes
+
+\ccNestedType{CurvePairAnalysis_2}{A model of
+\ccc{AlgebraicKernelCCPA_2::CurvePairAnalysis_2}, 
+for analysing a pair of curves
+defined as two analyses of type \ccc{CurveAnalysis_2}.}
+
+\ccNestedType{CurveAnalysis_2}{A model of
+\ccc{CurvePairAnalyis_2::CurveAnalysis_2}, for analysing single
+curves defined as bivariate polynomials of type \ccc{Polynomial_2}.}
+
+{\small TODO: Boundary: RealEmbeddable that is 'dense R'??}
+
 \end{ccRefConcept}
diff --git a/Algebraic_kernel_d/doc_tex/Curve_and_curve_pair_analysis_2_ref/Solution_1.tex b/Algebraic_kernel_d/doc_tex/Curve_and_curve_pair_analysis_2_ref/Solution_1.tex
index 17c21d1f5ba..8a263a107ef 100644
--- a/Algebraic_kernel_d/doc_tex/Curve_and_curve_pair_analysis_2_ref/Solution_1.tex
+++ b/Algebraic_kernel_d/doc_tex/Curve_and_curve_pair_analysis_2_ref/Solution_1.tex
@@ -6,11 +6,22 @@ The concept \ccc{Solution_1} is meant to store a coordinate ($x$, or $y$) of
 a point on curve. 
 
 \ccRefines
- \ccc{RealEmbeddable}
+ \ccc{DefaultConstructible, Assignable, LessThanComparable, CompactifiedNumber}
 
-Remark: Michael Hemmer proposed to extend the concept \ccc{RealEmbeddable} 
-with the notion of compactification (adding/dealing with $\pm\infty$). A
-valid model has to implement handling of infinity.
+The concept \ccc{CompactifiedNumber} has not been introduced so far. Michael
+Hemmer will introduce it as basic tool as soon as possible. For
+now we assume that a number of this type is either finite, and if not,
+we can ask, wether it represents $\pm\infty$.
+
+There are reasons why we introduce \ccc{Solution_1} over 
+\ccc{AlgebraicReal_1}\footnote{Not yet introduced so far}:
+\begin{itemize}
+\item It may suffices to use a simpler number type ($Q[\sqrt{2}]$).
+\item Keep concept minimal, \ccc{AlgebraicReal_1} will quite certainly
+contain more than needed here.
+\item \ccc{AlgebraicReal_1} won't be a refinement of \ccc{CompaticiedNumber}.
+\ccc{double} is.
+\end{itemize}
 
 \begin{ccAdvanced}
 
@@ -38,7 +49,7 @@ following methods:
   \ccPrecond{sol1 != s}
 }
 
-\ccMethod{std::pair<Boundary,Boundary> s1.approximate(int prec);}{
+\ccMethod{std::pair<Boundary,Boundary> approximate(int prec);}{
   Refines the representation to the given precision (binary digits 
   after point). Internally the precision can already be higher.
 } 
@@ -48,4 +59,11 @@ Maybe, we can have it orthogonal to the normal concept.
 
 \end{ccAdvanced}
 
+Remark: Note that this concept only deals with the interface to upper
+layers. There might be additional requirements for number types to 
+implement a model of this concept.
+
+\ccHasModels
+\ccc{double, NiX::Compactified< NiX::Algebraic_real >}
+
 \end{ccRefConcept}
diff --git a/Algebraic_kernel_d/doc_tex/Curve_and_curve_pair_analysis_2_ref/Solution_2.tex b/Algebraic_kernel_d/doc_tex/Curve_and_curve_pair_analysis_2_ref/Solution_2.tex
index 0eff4062fc4..99d02978ece 100644
--- a/Algebraic_kernel_d/doc_tex/Curve_and_curve_pair_analysis_2_ref/Solution_2.tex
+++ b/Algebraic_kernel_d/doc_tex/Curve_and_curve_pair_analysis_2_ref/Solution_2.tex
@@ -5,6 +5,18 @@
 The concept \ccc{Solution_2} is meant to store both coordinates of a point
 on a curve.
 
+There are reasons why we introduce \ccc{Solution_2} over 
+\ccc{AlgebraicReal_2}:
+\begin{itemize}
+\item \ccc{AlgebraicReal_2} seems to be pair of \ccc{AlgebraicReal_1}, while
+\ccc{Solution_2} has members to access \ccc{Solution_1} for $x$ and $y$. 
+With the name we state better to be more generic.
+\item Keep concept minimal, in contrast to \ccc{AlgebraicReal_2} that
+seems to be quite powerful.
+\item \ccc{Solution_2} is ``compactified'' (with the help of 
+\ccc{Solution_1}) in contrast to \ccc{AlgebraicReal_2}.
+\end{itemize}
+
 \ccCreationVariable{sol2}
 
 \ccAccessFunctions